Lyapunov-based robust optimal control for time-delay systems with application in milling process

In this paper, we propose an optimal delay-independent robust controller based on the Lyapunov-Krakovskii theorem for the milling process as a time-delay system in the presence of varying axial depth of cut and different parametric uncertainty, such as stiffness, damping, etc. Milling is widely used in the manufacturing processes for the production of complex-shaped workpieces with high accuracy. However, chatter is a self-excited vibration that may cause adverse effects, such as tool damage, poor surface quality, excessive noise, etc. The dynamic model of the milling process is considered a two-dimensional time-delay system. A nonlinear programming with linear matrix inequality constraints is solved in order to obtain the controller gain where the objective function is corresponding to the norm-2 of controller gain, and the constraints guarantee robust stability. Using the semi-discretization method, stability lobes are shown in both uncontrolled and controlled plants to illustrate the improvement of the stable region via the proposed controller. Bifurcation phenomena have been improved with this controller by postponing the adverse effects to the higher values of the axial depth of cut, reducing the amplitude of limit cycles, and changing the type of bifurcation. Finally, we will compare the proposed controller with an intelligent controller in order to show the efficiency of the proposed method. It is shown that the proposed controller has improved the integral absolute error index by about 3.65 times compared with the intelligent controller.